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A304447
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Coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(2*n).
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2
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1, 4, 40, 448, 5264, 63624, 783328, 9770240, 123040288, 1561033348, 19922193200, 255472920256, 3289122824000, 42488488508808, 550435283089088, 7148519205631488, 93038785849116736, 1213215382135324680, 15846906866928513736, 207302985358274247104
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) ~ c * d^n / sqrt(n), where d = 13.43567525239504624062504283058713960962824709850658926621911428148173077464... and c = 0.3323527904383991069791889982282236666403568774227549868882810268779...
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MATHEMATICA
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nmax = 20; Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^(2*n), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2/QPochhammer[x])^(2*n), {x, 0, n}], {n, 0, nmax}]
(* Calculation of constants {d, c}: *) eq = FindRoot[{QPochhammer[-1, r*s] == 2*Sqrt[s]*QPochhammer[r*s], (QPochhammer[ r*s]*(Log[r*s] - 2*Log[1 - r*s] - 2*QPolyGamma[0, 1, r*s])) / Log[r*s] - r*Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s] + 2*r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == 0}, {r, 1/12}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[((1 - r*s)*Log[r*s]^2*QPochhammer[r*s]) / (Pi*(2*r*s*(-1 + r*s) * Log[r*s]*(2*(Log[r*s] - 2*Log[1 - r*s] - 2*QPolyGamma[0, 1, r*s]) * Derivative[0, 1][QPochhammer][r*s, r*s] + r*Sqrt[s]*Log[r*s] * (-Derivative[0, 2][QPochhammer][-1, r*s] + 2*Sqrt[s]*Derivative[0, 2][QPochhammer][r*s, r*s])) + QPochhammer[ r*s]*(16*r*s*ArcTanh[1 - 2*r*s] + (1 - r*s)*Log[r*s]^2 - 8*Log[1 - r*s] + 4*(-1 + r*s)*Log[1 - r*s]^2 + 8*(-1 + r*s)*(1 + Log[1 - r*s])* QPolyGamma[0, 1, r*s] + 4*(-1 + r*s)*QPolyGamma[0, 1, r*s]^2 + 4*(-1 + r*s)*(QPolyGamma[1, 1, r*s] - 2*r*s*Log[r*s]*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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